Find the value of the measure of angle 𝐸 given that angle 𝐸 is an acute angle and cos of 𝐸 is 0.5201. Give your answer to the nearest second.
So we have a trigonometric equation here. cos of 𝐸 is equal to 0.5201. We’re going to need to solve this equation for 𝐸 to help us find the value of the measure of angle 𝐸. And since the question tells us to give our answer to the nearest second, that tells us two things. Firstly, we know we’re going to be working in degrees, minutes, and seconds, rather than, of course, radian measure. Secondly, we know that since we’re going to be rounding our answer, that implies that we’re going to be using a calculator to work this one out. So how do we solve a trigonometric equation of this form?
A really common mistake here is to think that, to solve this equation, we divide both sides by cos. But in fact, cos itself is an operation and the opposite operation, the inverse to finding the cos of something, is to find the inverse cos. And so we find the inverse cos of both sides of our equation. The inverse cos of cos of 𝐸 is just 𝐸, and so 𝐸 must be equal to the inverse cos of 0.5201. Let’s type this into our calculator. The inverse cos of 0.5201 is 58.66104 and so on degrees. Now, of course, we need to convert this to degrees, minutes, and seconds. And so we could hit the relevant button on our calculator. It looks a little something like this. Should we not have that functionality, though, we can work this out by hand.
We’re going to subtract the integer part of our number; we’re going to subtract 58. And so we get 0.6610 and so on. We know that when we’re working with degrees, minutes, and seconds we’re actually working in base 60. So we take this decimal part and we multiply it by 60. That gives us 39.6624 and so on. So that’s the minutes part of our answer. To find the seconds part, we repeat the process once again. We take away the integer part, leaving us with 0.66 and so on, and then we multiply that by 60. That’s 39.74, which is roughly 40 correct to the nearest second.
So the degrees part of our answer is 58, the minutes part is 39, and the seconds part is 40. This is indeed an acute angle as required. And so we find the measure of angle 𝐸 is 58 degrees, 39 minutes, and 40 seconds.